Problem 16 - Algebra 2 and Trigonometry

Question

In \(9-20,\) for each given function value, find \(a \cdot \sin 2 \theta, \mathbf{b} \cdot \cos 2 \theta, \mathbf{c} \cdot \tan 2 \theta, \mathbf{d}\) . the quadrant in which 2\(\theta\) lies. Show all work. \(\sin \theta=-\frac{\sqrt{2}}{3}\) in the third quadrant

Step-by-Step Solution

Answer

\( 2\theta \) is in the first quadrant; compute \( \sin 2\theta \), \( \cos 2\theta \), \( \tan 2\theta \) using double-angle identities.

Step 1: Determine the Reference Angle

Since \( \sin \theta = -\frac{\sqrt{2}}{3} \) and \( \theta \) is in the third quadrant, we identify that to find the reference angle, we need to consider the positive value, \( \sin \theta = \frac{\sqrt{2}}{3} \). Use the inverse sine function to find the reference angle associated with this sine value.

Step 2: Determine the Angle "\( \theta \)" in the Third Quadrant

The reference angle helps us determine the exact angle \( \theta \) in the third quadrant. In the unit circle, the third quadrant angles from \( 180^\circ \) to \( 270^\circ \). Therefore, \( \theta = 180^\circ + \text{reference angle} \).

Step 3: Find \( 2\theta \)

Once \( \theta \) is calculated, we double the angle to find \( 2\theta \). Calculate \( 2\theta = 2 \times \theta \).

Step 4: Identify the Quadrant of \( 2\theta \)

To determine the quadrant in which \( 2\theta \) lies, analyze the range of \( 2\theta \). Since \( \theta \) originally spans from \( 180^\circ \) to \( 270^\circ \) in the third quadrant, \( 2\theta \) would logically fall between \( 360^\circ \) to \( 540^\circ \). This corresponds to the first quadrant when considering angles within a standard \( 0^\circ \) to \( 360^\circ \) circular wrap around by subtracting 360 degrees if necessary.

Step 5: Calculate \( \sin 2\theta \), \( \cos 2\theta \), \( \tan 2\theta \)

Use the double angle identities to find \( \sin 2\theta \), \( \cos 2\theta \), and \( \tan 2\theta \):- \( \sin 2\theta = 2 \sin \theta \cos \theta \), requires calculation with \( \cos \theta = -\sqrt{1-\sin^2 \theta} \) due to the third quadrant.- \( \cos 2\theta = \cos^2 \theta - \sin^2 \theta \).- \( \tan 2\theta = \frac{2 \tan \theta}{1 - \tan^2 \theta} \), calculate \( \tan \theta = \frac{\sin \theta}{\cos \theta} \).

Step 6: Final Step: Consolidate Results

After computing the above values for \( \sin 2\theta \), \( \cos 2\theta \), and \( \tan 2\theta \), along with the quadrant, summarize all findings in a clear manner.

Key Concepts

Double Angle Identities

In trigonometry, double angle identities are vital for simplifying expressions involving trigonometric functions of doubled angles. These identities help us express functions like \( \sin 2\theta, \cos 2\theta\), and \( \tan 2\theta \) in terms of \( \sin \theta \) and \( \cos \theta \).

• \( \sin 2\theta = 2 \sin \theta \cos \theta \): This identity uses both sine and cosine of the single angle \( \theta \).
• \( \cos 2\theta = \cos^2 \theta - \sin^2 \theta \): This is derived from the Pythagorean identity, expressing cosine of doubled angle through sin and cos squared.
• \( \tan 2\theta = \frac{2 \tan \theta}{1 - \tan^2 \theta} \): Here, we relate the tangent of the doubled angle to tangent of \( \theta \).

These identities are not just theoretical—they are practical tools for solving equations and problems in trigonometry. They simplify calculations and help in determining the precise nature of angles.

Unit Circle

The unit circle is a fundamental concept in trigonometry where every angle has a corresponding point on a circle with a radius of one. This setup allows all trigonometric functions to be expressed as coordinates of points of the circle.The circle is usually centered at the origin of a coordinate system. Here are key points:

• Radius: Always 1 unit, which standardizes calculations across different angles.
• Coordinates: Each point on the circle can be represented as \( (\cos \theta, \sin \theta) \).
• 360 Degrees: Completing a full circle brings you back to the start point, making trigonometric functions periodic.

Using the unit circle, we can easily determine the sign and value of trigonometric functions in any quadrant, important for problems like identifying the quadrant of your resulting angle.

Reference Angle

The reference angle is the acute angle a given angle makes with the x-axis. Exploring reference angles helps us understand the behavior of trigonometric functions in different quadrants by simplifying angles back to the first quadrant.Key facts about reference angles:

• Acute Angles: All reference angles are acute, meaning between 0 and \(90^\circ\).
• Reflection: For every angle in the unit circle, the reference angle is essentially the angle's shadow on the x-axis, typically expressed in its simplest form—the first quadrant equivalent.
• Usage: Understanding reference angles allows for calculating trigonometric values using known values from the first quadrant.

They're used to map values both forward and backward across the quadrants, bridging knowledge from easier angles to more complex scenarios.

Third Quadrant

The third quadrant of the unit circle is a unique segment where both sine and cosine take negative values.This quadrant spans from \(180^\circ\) to \(270^\circ\) and has some specific characteristics:

• Sign Consistency: In this quadrant, both sine and cosine are negative, resulting in a positive tangent (as tan is a ratio of sine and cosine).
• Location: Angles here are those that continue from \(180^\circ\) to \(270^\circ\), basically representing the lower left part of the unit circle.
• Practicality: Many real-world problems use this quadrant to understand how angles reflect negative sine and cosine values.

Understanding the third quadrant is key to solving angles and evaluating which trigonometric functions will yield positive or negative results.